• Amir Mafi

      Articles written in Proceedings – Mathematical Sciences

    • A Generalization of the Finiteness Problem in Local Cohomology Modules

      Amir Mafi

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      Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring 𝑅 with non-zero identity and let 𝑁 be a weakly Laskerian 𝑅-module and 𝑀 be a finitely generated 𝑅-module. Let 𝑡 be a non-negative integer. It is shown that if $H^i_{\mathfrak{a}}(N)$ is a weakly Laskerian 𝑅-module for all $i < t$, then $\mathrm{Hom}_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(M, N))$ is weakly Laskerian 𝑅-module. Also, we prove that $\mathrm{Ext}^i_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(N))$ is weakly Laskerian 𝑅-module for all $i=0,1$. In particular, if $\mathrm{Supp}_R(H^i_{\mathfrak{a}}(N))$ is a finite set for all $i < t$, then $\mathrm{Ext}^i_R(R/\mathfrak{a},H^t_{\mathfrak{a}}(N))$ is weakly Laskerian 𝑅-module for all $i=0,1$.

    • Results on the Hilbert coefficients and reduction numbers

      AMIR MAFI DLER NADERI

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      Let $(R,\frak{m})$ be a $d$-dimensional Cohen--Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert coefficients. In addition, we also give some examples.

    • Linear resolutions and polymatroidal ideals

      AMIR MAFI DLER NADERI

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      Let $R = K[x_1,\ldots , x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog (Eur. J. Combin. 34 (2013) 752--763) conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) ${\rm height}(I) = n − 1$; (ii) $I$ contains at least $n − 3$ pure powers of the variables $x^d_1 ,\ldots , x^d_{n−3}$; (iii) $I$ is a monomial ideal in at most four variables.

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